17. Divergence, Curl and Potentials

c. The Curl Operator

1. Algebraic Definition of Curl

The cross product of two vectors, \(\vec u\times\vec v\), is the determinant of the matrix with \(\hat\imath\), \(\hat\jmath\) and \(\hat k\) on the first row, the vector \(\vec u\) on the second row and the vector \(\vec v\) on the third row.

The curl of a vector field, \(\vec\nabla\times\vec F\), is the cross product of the del operator and the vector field with the exception that when a differential operator "multiplies" a component of the vector field, it differentiates it.

\[ \vec u\times\vec v =\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} \] \[ \vec\nabla\times\vec F =\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ \partial_x & \partial_y & \partial_z \\ F_1 & F_2 & F_3 \end{vmatrix} \]

The curl (or rotation) of a vector field \(\vec F=\left\langle F_1, F_2, F_3 \right\rangle\), is the vector field \[\begin{aligned} \text{curl}\vec F &=\text{rot}\vec F =\vec\nabla\times\vec F =\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ \partial_x & \partial_y & \partial_z \\ F_1 & F_2 & F_3 \end{vmatrix} \\ &=\hat\imath(\partial_y F_3-\partial_z F_2) -\hat\jmath(\partial_x F_3-\partial_z F_1) +\hat k (\partial_x F_2-\partial_y F_1) \\ &=\left\langle \partial_y F_3-\partial_z F_2, \partial_z F_1-\partial_x F_3, \partial_x F_2-\partial_y F_1 \right\rangle \end{aligned}\] Be careful with the sign of the second component!
The name rotation and the notation \(\text{rot}\vec F\) are archaic and only appear in some old math and science articles.

Find the curl of \(\vec F=\left\langle x^2y,y^2+z^2,z^3-x^2z\right\rangle\).

We use the definition of the curl: \[\begin{aligned} \vec\nabla\times\vec F &=\begin{vmatrix} \; \; \hat\imath & \hat\jmath & \hat k \\ \; \; \partial_x & \partial_y & \partial_z \\ \; \; x^2y & y^2+z^2 & z^3-x^2z \end{vmatrix} \\ &=\hat\imath(0-2z)-\hat\jmath(-2xz-0)+\hat k(0-x^2) \\ &=\left\langle-2z,2xz,-x^2 \right\rangle \end{aligned}\] Be careful with the sign of the second component!

Fluid Velocity Interpretation

If the vector field is the velocity field of a fluid, \(\vec V\), then the curl of the velocity field, \(\vec\nabla\times\vec V\), measures of how much the fluid is rotating at each point. The direction of the curl is the axis of rotation. The magnitude of the curl is the rate at which the vector field is rotating. The direction of rotation is determined by the right hand rule. In particular, if you hold your right hand in a fist and point your thumb along the direction of the curl, then the fluid rotates in the direction of your fingers, i.e. counterclockwise as seen from the tip of your thumb. (See the next page.) If \(\vec\nabla\times\vec V=0\), we say the vector field is curl-free and the fluid is irrotational.

Compute the curl of the the velocity field \(\vec V=\left\langle xz-xy^2, x^2+yz^2, xyz^3 \right\rangle\).

\(\vec\nabla\times\vec V =\left\langle xz^3-2yz, x-yz^3, 2x+2xy \right\rangle\)

We use the definition of the curl: \[\begin{aligned} \vec\nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ \partial_x & \partial_y & \partial_z \\ xz-xy^2 & x^2+yz^2 & xyz^3 \end{vmatrix} \\ &=\hat\imath(xz^3-2yz)-\hat\jmath(yz^3-x)+\hat k(2x--2xy) \\ &=\left\langle xz^3-2yz, x-yz^3, 2x+2xy \right\rangle \end{aligned}\] Be careful with the sign of the second component!

Compute the curl of the the velocity field \(\vec V=\left\langle 3xy^2, 3x^2y, z^3 \right\rangle\).

\(\vec{\nabla}\times\vec V =\left\langle 0,0,0 \right\rangle\)

We use the definition of the curl: \[\begin{aligned} \nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ \partial_x & \partial_y & \partial_z \\ 3xy^2 & 3x^2y & z^3 \end{vmatrix} \\ &=\hat\imath(0-0)-\hat\jmath(0-0)+\hat k(6xy-6xy) \\ &=\left\langle 0,0,0 \right\rangle \end{aligned}\] So this velocity field is irrotational.

17. Divergence, Curl and Potentials

c. The Curl Operator

1. Algebraic Definition of Curl

Fluid Velocity Interpretation

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